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Communications on Pure & Applied Analysis

2011 , Volume 10 , Issue 6

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A decomposition theorem for $BV$ functions
Stefano Bianchini and  Daniela Tonon
2011, 10(6): 1549-1566 doi: 10.3934/cpaa.2011.10.1549 +[Abstract](42) +[PDF](417.9KB)
The Jordan decomposition states that a function $f: R\to R$ is of bounded variation if and only if it can be written as the difference of two monotone increasing functions.
In this paper we generalize this property to real valued $BV$ functions of many variables, extending naturally the concept of monotone function. Our result is an extension of a result obtained by Alberti, Bianchini and Crippa.
A counterexample is given which prevents further extensions.
Characterization of the value function of final state constrained control problems with BV trajectories
Ariela Briani and  Hasnaa Zidani
2011, 10(6): 1567-1587 doi: 10.3934/cpaa.2011.10.1567 +[Abstract](40) +[PDF](684.6KB)
This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories, involving time-measurable Hamiltonian. We study the characterization of the value function of this auxiliary problem and discuss its numerical approximations.
Vortex interaction dynamics in trapped Bose-Einstein condensates
Pedro J. Torres , R. Carretero-González , S. Middelkamp , P. Schmelcher , Dimitri J. Frantzeskakis and  P.G. Kevrekidis
2011, 10(6): 1589-1615 doi: 10.3934/cpaa.2011.10.1589 +[Abstract](31) +[PDF](3461.7KB)
Motivated by recent experiments studying the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates (BECs), we illustrate that such systems can be accurately described by ordinary differential equations (ODEs) incorporating the precession and interaction dynamics of vortices in harmonic traps. This dynamics is tackled in detail at the ODE level, both for the simpler case of equal charge vortices, and for the more complicated (yet also experimentally relevant) case of opposite charge vortices. In the former case, we identify the dynamics as being chiefly quasi-periodic (although potentially periodic), while in the latter, irregular dynamics may ensue when suitable external drive of the BEC cloud is also considered. Our analytical findings are corroborated by numerical computations of the reduced ODE system.
A generalization of $H$-measures and application on purely fractional scalar conservation laws
Darko Mitrovic and  Ivan Ivec
2011, 10(6): 1617-1627 doi: 10.3934/cpaa.2011.10.1617 +[Abstract](63) +[PDF](398.2KB)
We extend the notion of $H$-measures on test functions defined on $R^d\times P$, where $P\subset R^d$ is an arbitrary compact simply connected Lipschitz manifold such that there exists a family of regular nonintersecting curves issuing from the manifold and fibrating $R^d$. We introduce a concept of quasi-solutions to purely fractional scalar conservation laws and apply our extension of the $H$-measures to prove strong $L_{l o c}^1$ precompactness of such quasi-solutions.
Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain
Brahim Alouini
2011, 10(6): 1629-1643 doi: 10.3934/cpaa.2011.10.1629 +[Abstract](37) +[PDF](418.1KB)
We study the long-time behavior of the solutions to a nonlinear Schrödinger equation with a zero order dissipation and a quadratic potential when they are driven by an external force on a thin canal. We show that this behavior is described by a regular attractor which captures all the trajectories and have a finite Fractal dimension.
Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains
Antonio Capella
2011, 10(6): 1645-1662 doi: 10.3934/cpaa.2011.10.1645 +[Abstract](45) +[PDF](434.0KB)
We consider the nonlinear and nonlocal problem

$A_{1/2}u=|u|^{2^{\sharp}-2}u$ in $\Omega, \quad u=0$ on $\partial\Omega$

where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions, $\Omega$ is a bounded smooth domain in $R^n$, $n\ge 2$ and $2^{\sharp}=2n/(n-1)$ is the critical trace-Sobolev exponent. We assume that $\Omega$ is annular-shaped, i.e., there exist $R_2>R_1>0$ constants such that $\{ x \in R^n$ s.t. $R_1 < |x| < R_2 \}\subset\Omega$ and $0\notin\Omega$, and invariant under a group $\Gamma$ of orthogonal transformations of $R^n$ without fixed points. We establish the existence of positive and multiple sign changing solutions in the two following cases: if $R_1/R_2$ is arbitrary and the minimal $\Gamma$-orbit of $\Omega$ is large enough, or if $R_1/R_2$ is small enough and $\Gamma$ is arbitrary.

Unbounded solutions of the nonlocal heat equation
C. Brändle , E. Chasseigne and  Raúl Ferreira
2011, 10(6): 1663-1686 doi: 10.3934/cpaa.2011.10.1663 +[Abstract](34) +[PDF](461.7KB)
We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: $ u_t = J\ast u -u, $ where $J$ is a symmetric continuous probability density. Depending on the tail of $J$, we give a rather complete picture of the problem in optimal classes of data by: $(i)$ estimating the initial trace of (possibly unbounded) solutions; $(ii)$ showing existence and uniqueness results in a suitable class; $(iii)$ proving blow-up in finite time in the case of some critical growths; $(iv)$ giving explicit unbounded polynomial solutions.
Self-adjoint, globally defined Hamiltonian operators for systems with boundaries
Nuno Costa Dias , Andrea Posilicano and  João Nuno Prata
2011, 10(6): 1687-1706 doi: 10.3934/cpaa.2011.10.1687 +[Abstract](41) +[PDF](427.7KB)
For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(R^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(R^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-\Delta +V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+ B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.
Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations
Limei Dai
2011, 10(6): 1707-1714 doi: 10.3934/cpaa.2011.10.1707 +[Abstract](46) +[PDF](314.6KB)
In this paper, we use the Perron method to prove the existence of viscosity solutions with asymptotic behavior at infinity to fully nonlinear uniformly elliptic equations in $R^n$.
On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces
Lucas C. F. Ferreira and  Elder J. Villamizar-Roa
2011, 10(6): 1715-1732 doi: 10.3934/cpaa.2011.10.1715 +[Abstract](41) +[PDF](445.4KB)
We analyze the local well-posedness of the initial-boundary value problem for the heat equation with nonlinearity presenting a combined concave-convex structure and taking the initial data in weak-$L^{p}$ spaces. Moreover we give a new uniqueness class and obtain some results about the behavior of the solutions near $t=0^{+}.$
Blow-up rates of large solutions for semilinear elliptic equations
Zhijun Zhang and  Ling Mi
2011, 10(6): 1733-1745 doi: 10.3934/cpaa.2011.10.1733 +[Abstract](46) +[PDF](385.0KB)
In this paper we analyze the blow-up rates of large solutions to the semilinear elliptic problem $\Delta u =b(x)f(u), x\in \Omega, u|_{\partial \Omega} = +\infty,$ where $\Omega$ is a bounded domain with smooth boundary in $R^N$, $f$ is rapidly varying or normalised regularly varying with index $p$ ($p>1$) at infinity, and $b \in C^\alpha (\bar{\Omega})$ which is non-negative in $\Omega$ and positive near the boundary and may be vanishing on the boundary.
A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities
Bernd Kawohl and  Vasilii Kurta
2011, 10(6): 1747-1762 doi: 10.3934/cpaa.2011.10.1747 +[Abstract](41) +[PDF](417.4KB)
We compare entire weak solutions $u$ and $v$ of quasilinear partial differential inequalities on $R^n$ without any assumptions on their behaviour at infinity and show among other things, that they must coincide if they are ordered, i.e. if they satisfy $u\geq v$ in $R^n$. For the particular case that $v\equiv 0$ we recover some known Liouville type results. Model cases for the equations involve the $p$-Laplacian operator for $p\in[1,2]$ and the mean curvature operator.
Theory of the NS-$\overline{\omega}$ model: A complement to the NS-$\alpha$ model
W. Layton , Iuliana Stanculescu and  Catalin Trenchea
2011, 10(6): 1763-1777 doi: 10.3934/cpaa.2011.10.1763 +[Abstract](54) +[PDF](407.9KB)
We study a recent regularization of the Navier-Stokes equations, the NS-$\overline{\omega}$ model. This model has similarities to the NS-$\alpha$ model, but its structure is more amenable to be used as a basis for numerical simulations of turbulent flows. In this report we present the model and prove existence and uniqueness of strong solutions as well as convergence (modulo a subsequence) to a weak solution of the Navier-Stokes equations as the averaging radius decreases to zero. We then apply turbulence phenomenology to the model to obtain insight into its predictions.
Even solutions of the Toda system with prescribed asymptotic behavior
Yong Liu
2011, 10(6): 1779-1790 doi: 10.3934/cpaa.2011.10.1779 +[Abstract](48) +[PDF](336.4KB)
Every solution of the Toda system, describing the behavior of a finite number of mass points on the line, each one interacting with its neighbors, is asymptotically linear at infinity. We show the existence and uniqueness of even solution with suitable prescribed asymptotic behavior, by analyzing a system of algebraic equations derived from the relation between the slopes and the intercepts of the asymptotic lines.
A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems
D. Motreanu , Donal O'Regan and  Nikolaos S. Papageorgiou
2011, 10(6): 1791-1816 doi: 10.3934/cpaa.2011.10.1791 +[Abstract](43) +[PDF](502.0KB)
In this paper we present a framework which permits the unified treatment of the existence of multiple solutions for superlinear and sublinear Neumann problems. Using critical point theory, truncation techniques, the method of upper-lower solutions, Morse theory and the invariance properties of the negative gradient flow, we show that the problem can have seven nontrivial smooth solutions, four of which have constant sign and three are nodal.
Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$
Salvador Villegas
2011, 10(6): 1817-1821 doi: 10.3934/cpaa.2011.10.1817 +[Abstract](28) +[PDF](315.4KB)
We prove nonexistence of nonconstant global minimizers with limit at infinity of the semilinear elliptic equation $-\Delta u=f(u)$ in the whole $R^N$, where $f\in C^1(R)$ is a general nonlinearity and $N\geq 1$ is any dimension. As a corollary of this result, we establish nonexistence of nonconstant bounded radial global minimizers of the previous equation.
A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures
Irena Pawłow and  Wojciech M. Zajączkowski
2011, 10(6): 1823-1847 doi: 10.3934/cpaa.2011.10.1823 +[Abstract](47) +[PDF](420.2KB)
An initial-boundary-value problem for the sixth order Cahn-Hilliard type equation in 3-D is studied. The problem describes phase transition dynamics in ternary oil-water-surfactant systems. It is based on the Landau-Ginzburg theory proposed for such systems by G. Gompper et al. We prove that the problem under consideration is well posed in the sense that it admits a unique global smooth solution which depends continuously on the initial datum.

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